Thus, the pole-zero structures of a polynomial matrix at infinity are different from its “eigenvalue structure” at infinity defined through the reversal, but they are easily related through (REF ) and are completely equivalent to each other. From now on, we will make a clear distinction for any polynomial matrix \(P(\lambda )\)  of degree \(d\) : whenever we talk about its “eigenvalue structure at infinity”, we refer to the zero structure of \(\text{rev}_d P(\lambda )\)  at 0, and whenever we talk about its “pole or zero structures at infinity”, we refer to the pole or zero structures of \(P(1/\lambda )\)  at 0. Recall that such a distinction is not necessary at finite points. Moreover, we emphasize that a polynomial matrix of degree \(d>0\)  may or may not have an eigenvalue at infinity, may or may not have a zero at infinity, but always has a pole at infinity with largest partial multiplicity (or order) \(d\) . More on this topic can be found in [1]}. Finally, note that for pencils, i.e., polynomial matrices with degree 1, the definition of “eigenvalue structure at infinity” via reversals is equivalent to that coming from the Kronecker canonical form [2]}, and that the relation (REF ) was pointed out in [3]}.
